Application of Finite Univariate Gaussian Mixture Models for High-Frequency Financial Data Modelling

Risk management is one of the key factors in the financial markets in general and derivative pricing industry in specific. Estimation of risk in finance requires flexible models capable of giving the best fit to financial data, especially high-frequency data, to prevent under estimation or over-estimation of financial risk. The aim of this research is to use a flexible Gaussian mixture in modeling high-frequency financial data. The Expectation Maximization algorithm is used in estimating the parameters of the Gaussian mixture model. The research uses high-frequency financial data from Chinese Soy Beans Futures which were captured at a minute interval. The iterative process of the Expectation Maximization algorithm for a continuous probability density is shown and hence, for a two component Gaussian mixture. The generalization has been given for a \(\kappa\)-component mixture with probability wi where i is the i-th component. The flexible distributions have been applied to the data each time with increasing component starting with two components. Selection criterion is based on the BIC values which were found to ignore over-fitting. The BIC gradient were computed and used as selection criterion instead. We found that beyond five components, if we increase the number of components, the BIC gradient scores remains almost constant. Therefore, there is no gain in the mixture model by increasing the number of components. As a result, we conclude that the best model is the one with four components. Future research is however required to come up with the best model under BIC criterion with penalty for over-fitting as the model complexity increases.